Versor

In mathematics, a versor is a quaternion of norm one, also known as a unit quaternion. Each versor has the form

${\displaystyle u=\exp(a\mathbf {r} )=\cos a+\mathbf {r} \sin a,\quad \mathbf {r} ^{2}=-1,\quad a\in [0,\pi ],}$

where the r² = −1 condition means that r is an imaginary unit. There is a sphere of imaginary units in the quaternions. Note that the expression for a versor is just Euler's formula for the imaginary unit r. In case a = π/2 (a right angle), then ${\displaystyle u=\mathbf {r} }$, and it is called a right versor.

The mapping ${\displaystyle q\to u^{-1}qu}$ corresponds to 3-dimensional rotation, and has the angle 2a about the axis r in axis–angle representation.

The collection of versors, with quaternion multiplication, forms a group, and appears as a 3-sphere in the 4-dimensional quaternion algebra.